Calderón-Zygmund operators on compact Lie groups
type: Item
instance of: scholarly article
title: Calderón-Zygmund operators on compact Lie groups
zbMATH Open document ID: 0841.43019
DOI: 10.1007/BF02572330
published in: Mathematische Zeitschrift
publication date: 1994-12-21T00:00:00Z
full work available at URL: https://eudml.org/doc/174659
review text: Let \(G\) be a connected, simply connected, compact semisimple Lie group of dimension \(n\), let \(g\) be the Lie algebra of \(G\), and let \(d\) be a bi-invariant metric on \(G\) induced by the Killing form on the complexification of \(g\). Suppose \(0 < \alpha < n\) and \(r > 0\). A convolution kernel of type \((\alpha, r)\) is a function \(K\) on \(G\) which is of class \(C^r\) on \(G \backslash \{e\}\) and satisfies \[ \biggl |X^J \bigl( K(x) \bigr) \biggr |\leq C_J d(x,e)^{\alpha - n - |J |} \tag{1} \] for all \(J = (j_1, \dots, j_n)\) with \(|J |\leq r\) and \(x \neq e\), where \(X^J = X_1^{j_1} X_2^{j_2} \dots X_n^{j_n}\). \(X_1, \dots, X_n\) is an orthogonal basis of \(g,e\) is the identity of \(G\), and \(C_J\) is a constant independent of \(x \in G\). Define the Calderón-Zygmund operator \(T_K\) with kernel \(K\) by \[ (T_Kf) (x) = \int_G f(y) K(xy^{-1}) dy, \quad x \in G, \] for \(f \in L^1 (G)\). If \(r\) is a positive integer, then a kernel of type \((0,r)\) is a distribution \(K\) on \(G\) which is of class \(C^r\) on \(G \backslash \{e\}\), satisfies (1) with \(\alpha = 0\), and guarantees the boundedness of \(T_K\) on \(L^2 (G)\). -- The main result of the paper is devoted to the boundedness of \(T_K\) on the real Hardy spaces \(H^p (G)\), \(p > 0\). Theorem: Suppose \(r\) is a positive integer and \(0 \leq \alpha < n\). If \(K\) is a kernel of type \((\alpha, r)\) and \({\alpha \over n} < {1 \over p_1} = {1 \over p_2} + {\alpha \over n} < 1 + {r \over n}\), then \(T_K\) extends to a bounded operator from \(H^{p_1} (G)\) to \(H^{p_2} (G)\).
Mathematics Subject Classification ID: 43A80
Mathematics Subject Classification ID: 42B20
Mathematics Subject Classification ID: 42B30
zbMATH DE Number: 704926
zbMATH Keywords: atom
zbMATH Keywords: Lie group
zbMATH Keywords: Lie algebra
zbMATH Keywords: Killing form
zbMATH Keywords: complexification
zbMATH Keywords: convolution kernel
zbMATH Keywords: Calderón-Zygmund operator
zbMATH Keywords: Hardy spaces
zbMATH Keywords: bounded operator
Wikidata QID: Q115391975
author: Dashan Fan
reviewed by: Yuri I. Karlovich
MaRDI profile type: Publication
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OpenAlex ID: W1963658695
description: scientific article; zbMATH DE number 704926